
theorem Th2:
  for p1,p2,p being Point of TOP-REAL 2,a being Real st p in LSeg(
  p1,p2) & p1`1<=a & p2`1<=a holds p`1<=a
proof
  let p1,p2,p be Point of TOP-REAL 2,a be Real;
  assume that
A1: p in LSeg(p1,p2) and
A2: p1`1<=a and
A3: p2`1<=a;
  consider r being Real such that
A4: p = (1-r)*p1+r*p2 and
A5: 0<=r and
A6: r<=1 by A1;
A7: p`1 = ((1-r)*p1)`1+(r*p2)`1 by A4,TOPREAL3:2
    .= ((1-r)*p1)`1+r*p2`1 by TOPREAL3:4
    .= (1-r)*p1`1+r*p2`1 by TOPREAL3:4;
  1-r>=0 by A6,XREAL_1:48;
  then
A8: (1-r)*p1`1<=(1-r)*a by A2,XREAL_1:64;
A9: (1-r)*a+r*a= a;
  r*p2`1<=r*a by A3,A5,XREAL_1:64;
  hence thesis by A7,A8,A9,XREAL_1:7;
end;
